Function gradient piezoelectric material optimization method based on physical information network and deep regression

Abstract

The invention discloses a functional gradient piezoelectric material optimization method based on a physical information network and deep regression. The method comprises the following steps of: 1, constructing a physical information neural network embedded with a piezoelectric constitutive relation; 2, developing a deep regression network based on symbolic regression, wherein the deep regression network is used for converting the piezoelectric phase volume fraction distribution predicted by the neural network into an analyzable mathematical expression; step 3, carrying out joint optimization on the two networks by adopting an alternate iteration strategy; under the condition that parameters of the deep regression network are kept unchanged, a physical information neural network is trained preferentially to meet the precision requirements of a displacement field, a stress field, potential distribution and a piezoelectric phase volume fraction; and then fixing the physical information network, optimizing the expression generation capability of the deep regression network, and finally analyzing the piezoelectric phase volume fraction expression. According to the method, the physical information neural network and the depth symbol regression technology are fused, so that multi-physical field collaborative optimization of component distribution of the functionally graded piezoelectric material is realized.

Description

Technical Field

[0001] The present invention relates to the field of smart material technology, and in particular to a method for optimizing functional gradient piezoelectric materials based on physical information networks and deep regression. Background Art

[0002] Piezoelectric materials have unique force-electric coupling characteristics and can realize the mutual conversion between mechanical energy and electrical energy. Therefore, they are widely used in devices such as sensors, transducers and actuators. At the same time, functionally gradient materials can optimize stress distribution through the spatial continuous change of material composition, effectively alleviate the stress concentration problem caused by sudden changes in composition of traditional composite materials, thereby significantly improving the reliability and service life of the structure. By applying the design concept of functionally gradient materials to the piezoelectric material system, the gradient distribution regulation of the components of functionally gradient piezoelectric materials along the spatial direction can be achieved, thereby achieving the optimal configuration of piezoelectric constants and dielectric properties in the spatial dimension, while ensuring that the material maintains excellent mechanical properties, and significantly improving its force-electric coupling conversion efficiency. This functionally gradient piezoelectric material shows broad application prospects in energy harvesting, electronic information, robotics and other fields.

[0003] However, the performance regulation of functionally gradient piezoelectric materials faces challenges. In the design process of functionally gradient piezoelectric materials, it is necessary to ensure that the gradient change of piezoelectric properties meets the design requirements, maintain the stability of dielectric properties, and also consider the continuity of mechanical properties. The complexity of this multi-objective collaborative optimization makes traditional design methods difficult to cope with. The current mainstream "finite element simulation + experimental verification" method has obvious limitations. When analyzing the mechanical-electrical coupling behavior of functionally gradient piezoelectric structures, it is necessary to solve the control equations at the same time, which is not only computationally intensive, but also when dealing with complex boundaries, the prediction accuracy of traditional numerical methods will drop significantly. Although data-driven methods based on machine learning provide new approaches for modeling complex materials, their dependence on high-quality training data sets seriously restricts the effectiveness of practical applications.

[0004] Physical information neural network provides a solution to this problem. This method directly embeds the constitutive equation of functionally graded piezoelectric materials into the neural network architecture and uses automatic differentiation technology to accurately solve the force-electric coupling behavior. Compared with traditional methods, this modeling method based on physical constraints not only reduces the dependence on experimental data, but also significantly improves the prediction accuracy in areas with high gradient changes. It is worth noting that this method still has certain shortcomings: its output results cannot provide clear information on the volume fraction distribution of each phase of the functionally graded piezoelectric material, making it difficult to directly guide the actual manufacturing process. This limitation is particularly prominent in the forming process of complex-shaped components.

[0005] Therefore, developing optimization design methods that simultaneously meet the constraints of physical laws and the requirement for analyzable output has become a key scientific challenge urgently needed in the research of functionally graded piezoelectric materials. Solving this challenge will enable the precise and controllable design of material properties, providing an important theoretical foundation and technical support for the development of a new generation of high-performance smart materials. Summary of the Invention

[0006] In order to overcome the defects of the above-mentioned existing technologies, the purpose of the present invention is to provide a functional gradient piezoelectric material optimization method based on physical information network and deep regression. This method realizes multi-physical field collaborative optimization of the component distribution of functional gradient piezoelectric materials by integrating physical information neural network and deep symbolic regression technology.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is:

[0008] The functional gradient piezoelectric material optimization method based on physical information network and deep regression includes the following steps:

[0009] Step 1: Construct a physical information neural network embedded with the piezoelectric constitutive relation;

[0010] The physical information neural network takes spatial coordinates and material parameters as input variables, and outputs key parameters such as displacement field, stress field, electric potential distribution, and piezoelectric phase volume fraction. Automatic differentiation technology is used to accurately calculate the stress field and electric displacement field. During the network training process, the mechanical equilibrium equations, electrical control equations, boundary conditions, and physical constraints of the optimization target are integrated into the loss function to ensure that the network prediction results strictly conform to the mechanical-electrical coupling characteristics of the piezoelectric material and achieve the optimization target.

[0011] It is worth noting that the network loss function is customizable and the optimization objective can be adjusted according to application requirements: structural sensors prioritize stress uniformity, while energy harvesters focus on output voltage optimization;

[0012] Step 2: Develop a deep regression network based on symbolic regression to convert the piezoelectric phase volume fraction distribution predicted by the neural network into an analyzable mathematical expression;

[0013] The deep regression network uses a reinforcement learning strategy to automatically search for the optimal expression form from a symbolic library containing basic operations, transcendental functions, and piezoelectric characteristic parameters. By introducing a reward mechanism that balances fitting accuracy and expression complexity, the resulting mathematical model is ensured to accurately reflect physical laws while also having good engineering feasibility. The innovation of this step lies in the transformation from the network's "black box" prediction to explicit function expression, providing direct guidance for subsequent material preparation.

[0014] Step 3: Use alternating iterative strategy to jointly optimize the two networks;

[0015] While keeping the parameters of the deep regression network unchanged, the physical information neural network is trained first to meet the accuracy requirements of the displacement field, stress field, electric potential distribution, and piezoelectric phase volume fraction. Then, the physical information network is fixed and the expression generation capability of the symbolic regression network is optimized to finally resolve the piezoelectric phase volume fraction expression.

[0016] Through this collaborative optimization mechanism, a piezoelectric phase distribution function was ultimately obtained that was both physically consistent and readily applicable in engineering applications. To verify the reliability of the design results, the Voigt homogenization method was used to evaluate equivalent material properties, ensuring that the gradient distribution met practical application requirements. This method overcomes the limitations of traditional optimization techniques in terms of accuracy and practicality, providing a new paradigm for the intelligent design of functionally graded piezoelectric materials.

[0017] The step 1 is specifically as follows:

[0018] Build a fully connected physical information neural network W net , the input is the constitutive parameters of the functionally graded piezoelectric material (such as the elastic modulus coefficient c ijkl , piezoelectric constant e ijk , dielectric constant κ ij ) and structural coordinate position points (x, y, z), the output is the displacement distribution (u, v, w), the potential distribution φ and the volume fraction function V A ,Right now

[0019] {(u,v,w) pred ,φ pred (x,y,z),V Apred (x,y,z)}=W net (x,y,z;ω,b) (1)

[0020] where u pred (x,y,z),v pred (x,y,z),w pred (x,y,z),φ pred (x,y,z) and V Apred (x, y, z) are the displacement distribution, potential distribution, and volume fraction function predicted based on the coordinate position (x, y, z), ω represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network. The network realizes the nonlinear transformation from the input space to the multi-physical field response through the parameterized mapping of the weight ω and the bias b.

[0021] The constitutive parameters of the functionally graded piezoelectric material are expressed as a function of volume fraction, and the equivalent material parameters are calculated by a homogenization model;

[0022] The functionally graded piezoelectric material is assumed to consist of component A (such as piezoelectric material PZT-4) and component B (such as piezoelectric material PZT-5H), and the volume fractions of component A and component B are V A (x,y,z) and V B (x,y,z), satisfying V A (x,y,z)+V B (x, y, z) = 1, using the Voigt mixing model, the equivalent material parameters are expressed as

[0023]

[0024] in and are the material parameters of the piezoelectric material PZT-4 (PZT-5H);

[0025] Based on the displacement and potential distribution output by the network, the strain and electric field intensity of the functionally graded piezoelectric structure are obtained by using the automatic differentiation technique.

[0026]

[0027] E i =-φ ,i (4)

[0028] where ε ij is the strain, u i is the displacement component, E i is the electric field intensity component, φ is the electric potential component, and the subscript “,” indicates the derivative.

[0029] Combined material equivalent parameter c ijkl 、e ijk , κ ij and the above strain ε ij and the electric field strength E i , using the constitutive equation, the stress and electric displacement components of the functionally graded piezoelectric structure are obtained as

[0030]

[0031] where σ ij is the stress component, D i is the electric displacement component.

[0032] The control equation of the equilibrium equation of the functionally graded piezoelectric structure is composed of the mechanical equilibrium equation and the electrical Gaussian law, namely:

[0033]

[0034] in is the body force component;

[0035] In the definition of the boundary Γ of the functionally graded piezoelectric structure, there are two boundary conditions: mechanical or electrical, namely

[0036]

[0037] in For a given displacement, Γ u is the displacement boundary of the piezoelectric structure, For a given surface tension, Γ σ is the stress boundary of the piezoelectric structure.

[0038] The electrical boundary conditions are

[0039]

[0040] in For a given electric displacement, Γ φ is the potential boundary of the piezoelectric structure, For a given charge, Γ q is the electric displacement boundary.

[0041] By substituting the constitutive equation into the equilibrium equation, the control equation with displacement and potential as the basic unknowns is established. Combined with the boundary conditions, the displacement, stress distribution and potential distribution of the functionally graded piezoelectric structure are obtained.

[0042] For functionally graded piezoelectric structures, the optimization objective is defined as

[0043]

[0044] in

[0045]

[0046] Or the optimization objective is defined as

[0047] φ M =max(φ output (x,y,z)) (11)

[0048] The network training process adopts a multi-objective optimization strategy. The total loss function Loss consists of three parts: the mean square error Loss of the balance equation f , boundary condition mean square error Loss b And the objective function mean square error Loss M ,Right now

[0049] Loss=Loss f +Loss b +Loss M (12)

[0050] Mean square error Loss of functional gradient structural equilibrium equationf

[0051]

[0052] The mean square error of the boundary conditions of the functionally graded piezoelectric structure is Loss b for

[0053]

[0054] Objective function mean square error Loss M for

[0055] Loss M =σ M or φ M (13c)

[0056] The residual term of the equilibrium equation is calculated through automatic differentiation, ensuring strict adherence to physical laws. The boundary condition term forces the network's predictions to meet given constraints. Finally, the objective function term guides the optimization process toward the desired performance indicator. This multi-loss function construction allows the network to balance the correctness of physical laws with practical engineering applications.

[0057] The step 2 is specifically as follows:

[0058] A deep regression network was constructed, which used a symbolic modeling method to establish an explicit mathematical relationship between spatial coordinates and the volume fraction of the piezoelectric phase. First, an expression generator based on a binary tree structure was established, with leaf nodes set as input variables (spatial coordinates x, y, z and parameters to be optimized c1-c n ), non-leaf nodes are configured as operator nodes (including the following basic arithmetic operators), namely

[0059]

[0060] In the initial stage, the Monte Carlo method is used to randomly generate 100 groups of initial expression trees. The depth of each expression tree is controlled at 5-8 layers to ensure moderate complexity.

[0061] The network optimization adopts a two-stage hybrid algorithm: in the first stage, a genetic algorithm is implemented to select and retain the expressions with the top 30% fitness; in the second stage, a gradient descent optimization is performed to fine-tune the leaf node parameters using an adaptive learning rate; after each iteration, the candidate expressions are double-evaluated: the normalized fitness index R = 1 / (1 + NRMSE) based on the output data of the physical information network is calculated, where NRMSE is the normalized root mean square error, which is defined as the root mean square of the deviation between the predicted value and the target value normalized by the data variance.

[0062]

[0063] Where S is the variance of the volume fraction data set calculated by the target volume fraction function, V A (x,y,z) and V A * (x, y, z) are the volume fraction function expressions predicted by the physical information network and the deep regression network, respectively.

[0064] During the expression screening phase, a multi-objective optimization mechanism is established, with three key constraints set: fitness index R ≥ 0.93, total number of nodes ≤ 35, and function depth ≤ 7. Expressions that pass the initial screening are verified for physical rationality, including dimensional consistency checks (forcing all operations to remain dimensionless or of the correct dimension) and boundary condition verification. Finally, the output phase generates the optimal solution set for the expression, providing 5-8 candidate expressions that achieve the best balance between accuracy and complexity.

[0065] The step three is specifically as follows:

[0066] The intelligent optimization of functionally graded piezoelectric materials is achieved through the collaborative work of physical information neural networks and deep regression networks. Two independent networks are configured in the initialization phase:

[0067] The physical information neural network adopts a 4-layer fully connected architecture. The input layer receives spatial coordinates and process parameters, and the output layer generates displacement fields, electric potential fields, and constitutive parameters. The network embeds the piezoelectric constitutive equation as a hard constraint.

[0068] The deep regression network is constructed as a symbolic regression network, which expresses the component distribution function based on a binary tree structure, and the node library contains basic operations.

[0069] The training process implements an intelligent alternating optimization strategy:

[0070] First, the parameters of the deep regression network are fixed, and the adaptive momentum optimizer is used to train the physical information neural network. The loss function includes the mechanical balance term, the electrical Gaussian law term, and the boundary conditions. The automatic differentiation technique is used to accurately calculate the stress field and the electric displacement field gradient. Then, the parameters of the physical information neural network are frozen, and the hybrid optimization algorithm of the deep regression network is enabled. Combining the genetic algorithm and the gradient descent method, the two indicators are calculated in real time during the optimization process: the goodness of fit R 2 (target > 0.99) and expression complexity (number of nodes ≤ 30);

[0071] The system sets a dynamic termination mechanism: monitoring the composite loss value of the physical information neural network (threshold 1×10 -5) and the normalized fitness R (threshold 0.99) of the deep regression network. Early stopping is triggered when both indicators meet the target for three consecutive rounds. After multiple rounds of iterative optimization, the system ultimately outputs an explicit mathematical expression with clear physical meaning. This expression is directly used to guide the setting of material preparation process parameters. Based on the optimized component distribution, the system also uses a Voigt homogenization calculation module to quickly and accurately predict the equivalent mechanical, electrical, and piezoelectric properties of the material, providing a reliable basis for engineering decision-making.

[0072] Beneficial effects of the present invention:

[0073] The present invention proposes a new paradigm for the optimization design of functional gradient piezoelectric materials that integrates physical mechanisms and data intelligence. This method creatively integrates the physical information neural network with the symbolic regression network in depth to construct an intelligent optimization system with autonomous knowledge discovery capabilities. The core advantage of this system is that it can automatically establish an accurate mapping relationship from the distribution of microscopic components to the macroscopic force-electric coupling performance, breaking through the reliance of traditional design methods on manual experience. By strictly embedding the piezoelectric constitutive equation, equilibrium equation and optimization target through the physical information network, it is ensured that the optimization process always follows the basic physical laws while achieving the optimization target; while the deep regression network uses symbolic expression to convert the implicit physical relationship into a resolvable mathematical formula. This dual-network architecture can not only accurately capture the complex nonlinear relationship between material components and multi-physical field properties (including elastic modulus coefficient, piezoelectric constant and dielectric constant, etc.), but more importantly, it can discover innovative gradient distribution laws that cannot be described by traditional empirical formulas, opening up new paths for the development of functional gradient piezoelectric materials with special properties.

[0074] In actual engineering applications, this method has shown significant technical advantages. The design of functionally gradient piezoelectric materials often requires seeking a balance between multiple objectives such as mechanical properties, electrical properties and preparation processes. Traditional methods require a lot of time for numerical simulation and experimental verification. The intelligent optimization system of the present invention realizes the automation and intelligence of the design process through a dual-network collaborative working mechanism. The physical information network ensures that all design schemes satisfy the basic force-electric coupling equation; and the deep regression network converts the optimization results into mathematical expressions that can be directly used for engineering implementation. This collaborative design model not only greatly improves design efficiency and shortens the traditional design cycle of several weeks to several days, but more importantly, it can explore design spaces that are difficult to reach with traditional methods, and discover more optimal material component distribution schemes, providing strong technical support for the engineering application of functionally gradient piezoelectric materials. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] Figure 1 Schematic diagram of the functionally graded piezoelectric material and the functionally graded piezoelectric hollow cylinder structure, which is a mixture of piezoelectric material PZT-4 (component A) and piezoelectric material PZT-5H (component B).

[0076] Figure 2 Schematic diagram of the process flow for optimizing the design of functionally graded piezoelectric hollow cylindrical structures based on deep regression and physical information networks.

[0077] Figure 3 Schematic diagram of the mechanical response of the functionally graded piezoelectric hollow cylinder based on its optimized design. (a) Displacement distribution, (b) radial stress distribution, (c) electric potential distribution, and (d) von Mises stress distribution. DETAILED DESCRIPTION

[0078] The present invention will be described in further detail below with reference to the accompanying drawings.

[0079] In the optimization design of functionally graded piezoelectric structures, the optimization goal is usually to minimize stress distribution or maximize output voltage, and then use known physical field information to infer the optimal material distribution to meet given performance indicators. The functionally graded material is assumed to be a composite of two piezoelectric ceramics (PZT-4 and PZT-5H), and the performance optimization is achieved by determining the optimal volume fraction distribution function. A (r) and V B (r) represents the volume fraction of component A (PZT-4) and component B (PZT-5H) along the radial direction, satisfying V A (r)+V B (r) = 1. The optimization goal is to determine the optimal volume fraction function V B (r), maximizing the output voltage of the functionally graded piezoelectric hollow cylinder while optimizing the stress distribution.

[0080] To complete this process, the distribution of material properties needs to be estimated based on a volume fraction homogenization scheme. The stress and voltage distributions are then obtained so that they meet the optimization objectives.

[0081] The present invention will be further described in detail below with reference to the embodiments.

[0082] The present invention is based on a functional gradient piezoelectric structure optimization design method based on physical information network and deep regression, and realizes an artificial intelligence neural network automated modeling scheme for the optimization design of functional gradient materials. Figure 1 As shown in the figure, consider a hollow cylinder made of radially polarized functionally graded piezoelectric ceramic material, whose thickness in the radial direction is R o -R i , where R i and R oare the inner and outer radii of the hollow cylinder, respectively. For ease of analysis, it is assumed that the functionally graded piezoelectric cylinder is subjected to axisymmetric mechanical loads and electric fields on its surface. Using a cylindrical polar coordinate system (r, θ, z), the z-axis is the longitudinal axis of the piezoelectric cylinder. Assuming the hollow cylinder is sufficiently long, the end effects of the structure can be neglected. In other words, all variables are independent of z and θ.

[0083] The following steps describe the present invention in detail. Figure 2 As shown:

[0084] Step 1: Construction of a physical information network to predict the coordinate position r and displacement distribution u(r), potential φ(r) and volume fraction function V of the functionally graded piezoelectric hollow cylinder A (r) mapping relationship.

[0085] First, establish a fully connected neural network W net (ω,b) Prediction of displacement distribution u(r), electric potential φ(r) and volume fraction function V A (r), where the neural network input is the coordinate position r and material parameters, and the output is the displacement distribution u(r), the potential φ(r) and the volume fraction function V A (r), as attached Figure 2 shown

[0086] {u pred (r),φ pred (r),V Apred (r)}=W net (r;ω,b) (1)

[0087] Among them, u pred 、φ pred and V Apred are the displacement distribution, potential and volume fraction predicted based on the coordinate position r, W net represents a fully connected neural network, ω represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network.

[0088] Based on elasticity theory, calculate different volume fraction functions V A (r) The corresponding Young's modulus under the expression. Taking the Voigt homogenization scheme as an example, different volume fraction functions V A The calculation formulas for the corresponding Young's modulus and Poisson's ratio under the expression (r) are as follows

[0089]

[0090] where c ij 、e ij and κ 33 are the elastic modulus, piezoelectric constant and dielectric constant, respectively.

[0091] Using the automatic differentiation calculation capability, the strain and electric field strength of the functionally graded piezoelectric hollow cylinder are obtained by displacement and potential calculation.

[0092]

[0093] Where u is the displacement in the r direction, ε rr ,ε θθ are the radial and hoop strain components, E r is the electric field intensity component.

[0094] Further combined with the constitutive equation, the stress and electric displacement prediction of the functionally graded piezoelectric hollow cylinder are calculated.

[0095] σ rr (r) = c 33 (r)ε rr (r)+c 13 (r)ε θθ (r)-e 33 (r)E r (r),

[0096] σ θθ (r) = c 13 (r)ε rr (r)+c 11 (r)ε θθ (r)-e 31 (r)E r (r), (5)

[0097] D r (r) = e 33 (r)ε rr (r)+e 33 (r)ε θθ (r)+κ 33 (r)E r (r)

[0098] The equilibrium equation of the functionally graded piezoelectric hollow cylindrical structure is:

[0099]

[0100] For a functionally graded piezoelectric hollow cylindrical structure, consider the following boundary conditions: the inner and outer surfaces of the functionally graded piezoelectric structure are subjected to uniformly distributed pressures p and q, or electric potentials V and 0, respectively. In this case, the boundary conditions can be expressed as follows:

[0101] sensor

[0102] σ rr (R i )=-p,σ rr (Ro )=-q,φ r (R i )=0,φ r (R o )=0(7a)

[0103] Drive

[0104] σ rr (R i )=0,σ rr (R o )=0,φ r (R i )=V,φ r (R o )=0(7b)

[0105] The optimization objective is defined as minimizing the von Mises stress, i.e.

[0106] σ M =min(maxσ von ) (9)

[0107] in

[0108]

[0109] The goal is to minimize σ M distribution.

[0110] And maximize the output voltage (potential difference), that is

[0111] φ M =Δφ=φ(R o )-φ(R i ) (11) The goal is to maximize φ M .

[0112] The displacement, potential and stress prediction of functionally graded piezoelectric hollow cylinders based on physical information networks have three constraints in the network training process.

[0113] Constraint 1 (control equation): ensure that the model follows the basic physical equations of functionally graded piezoelectric hollow cylinders;

[0114] Constraint 2: (Boundary conditions) Ensure that the predicted displacement field, stress field, and electric potential meet the boundary conditions of the functionally graded piezoelectric hollow cylinder;

[0115] Constraint 3: (Optimization objective) ensures that the optimization objective is achieved under the predicted volume fraction function. Through the combined effect of these three constraints, accurate prediction of functionally graded piezoelectric hollow cylinders can be achieved.

[0116] That is, the total loss function Loss of the physical information network includes the following parts: the mean square error Lossf of the equilibrium equation; the mean square error of the boundary condition and Loss b3 And the objective function mean square error Loss M ,Right now

[0117] Loss=Loss f +Loss b1 +Loss b2 +Loss M (12)

[0118] The mean square error Lossf of the equilibrium equation of the functionally graded piezoelectric hollow cylinder is:

[0119]

[0120] where N f is the number of sampling points of the residual term, r i are uniformly / randomly distributed internal coordinate points.

[0121] For the sensor, the residual terms of the mechanical and electrical boundary conditions of the functionally graded piezoelectric hollow cylinder are:

[0122]

[0123] Due to the axial symmetry, no hoop stress sampling is required, which means that only one sampling point is required at each of the inner and outer boundaries r = Ri and Ro.

[0124] For the actuator, the residual terms of the mechanical and electrical boundary conditions of the functionally graded piezoelectric hollow cylinder are:

[0125]

[0126] For von Mises stress minimization, the objective function is the mean square error Loss M1 for

[0127]

[0128] in

[0129]

[0130] For maximizing the output voltage (potential difference between the inner and outer surfaces), the objective function is the mean square error Loss M2 for

[0131]

[0132] It should be noted that the negative sign is because the loss function needs to be minimized, while the goal is to maximize it.

[0133] Objective function mean square error Loss M for

[0134]

[0135] Step 2: Building a deep regression network to generate a volume fraction function expression for optimizing the functionally graded piezoelectric hollow cylinder;

[0136] Build as attached Figure 2 The deep regression network shown in the figure analyzes the volume fraction function V represented by the coordinate point (r) of the functionally graded piezoelectric hollow cylinder. A * (r)Expression.

[0137] This network uses an innovative tree-structured representation to intelligently generate mathematical expressions. The network architecture maps each mathematical symbol to a binary tree node, where leaf nodes correspond to spatial coordinates (r) and constants to be optimized (e.g., k, m), while internal nodes represent various mathematical operators (including basic arithmetic operations such as +, -, ×, and / , trigonometric functions such as sin, cos, and tan, and logarithmic exponentials such as log and exp). During expression construction, a strict prefix notation (Polish notation) is used to convert complex mathematical relationships into one-dimensional symbol sequences. This representation not only eliminates redundant parentheses but also facilitates serialization within recurrent neural networks. To improve computational efficiency and expression rationality, the system imposes multiple intelligent constraints: limiting the maximum expression length to avoid overcomplication; controlling the nesting depth of trigonometric functions to prevent unrealizable oscillations; and prohibiting self-nesting of exponential / logarithmic operations and invalid inverse operations to ensure mathematical rationality. These carefully designed constraints enable the network to accurately capture the spatial variations in the distribution of piezoelectric material components while maintaining the simplicity of the expression. The final output of the explicit mathematical expression not only has a clear physical meaning, but can also be directly used to guide the setting of material preparation process parameters, achieving a perfect transformation from numerical optimization results to feasible engineering solutions.

[0138] When evaluating each candidate solution V A * (r), the deep regression network is calculated using the fitting quality reward R = 1 / (1 + NRMSE), where NRMSE is the root mean square error normalized by the deviation of the target S.

[0139]

[0140] Where S is the variance of the obtained volume fraction dataset, and V A (r) and V A *(r) are the volume fraction function expressions predicted by the physical information network and the deep regression network, respectively.

[0141] Step 3: Neural network training to find the optimal volume fraction function expression;

[0142] Based on the physical information network in step 1 and the deep regression network in step 2, joint training is performed and the training effect is ensured by setting a double convergence criterion. When the loss function of the physical information network is reduced to 10 -5 When the goodness-of-fit index R of the deep regression network exceeds 99%, the system automatically terminates the training process. This intelligent termination mechanism ensures that the model accurately follows the laws of physics and the accuracy of mathematical expressions, while avoiding unnecessary consumption of computing resources.

[0143] During the specific training process, the system adopts an alternating iterative optimization method to give full play to the complementary advantages of the two types of networks. First, the physical information network is optimized, and by minimizing the composite loss function containing multi-physical field constraints, the network accurately grasps the electromechanical coupling law of the piezoelectric material; then the parameters of the physical information network are fixed, and the deep regression network is optimized to improve its mathematical representation ability of the component distribution law. This alternating training strategy enables the two networks to promote each other and co-evolve through multiple rounds of iterations, ultimately achieving a steady improvement in overall performance. It is worth noting that each iteration will verify and adjust the previous optimization results to form a closed-loop optimization system, thereby obtaining the optimal solution with both physical accuracy and engineering practicality while ensuring computational efficiency.

[0144] After the above steps, the deep regression network finally outputs a volume fraction function V with clear physical meaning. A * (r) An explicit mathematical expression fully characterizes the optimal component distribution of functionally graded piezoelectric materials. Notably, all key parameters in this expression are determined by autonomous network optimization, without the need for human intervention. To verify the reliability of the results, the calculated expression can be compared with simulation data from traditional numerical methods (such as finite element analysis) to ensure that it meets engineering accuracy requirements.

[0145] In summary, the innovation of this method lies in its organic integration of physical mechanism modeling and data-driven optimization. Through the collaborative work of the two networks, a resolvable volume fraction function expression is directly output. This processing approach not only achieves the intelligent conversion of complex data into a concise mathematical model, but also simultaneously determines all the unknown parameters in the model, significantly simplifying the tedious parameter calibration steps in the traditional optimization design process. The entire optimization process forms a complete closed loop from physical modeling to mathematical expression.

[0146] Specifically, the physical information network first establishes a precise mapping from spatial coordinates to material properties, laying the physical foundation for subsequent analysis. Building on this foundation, the deep regression network uses symbolic regression techniques to transform implicit distribution patterns into explicit mathematical expressions. These two networks each perform their respective functions while working in tandem: the former ensures that optimization results conform to physical laws, while the latter provides mathematical tools that can be directly applied in engineering implementation. Together, they enable efficient optimization design of functionally graded piezoelectric materials.

[0147] Example:

[0148] The volume fraction expression of the intelligent material optimization design of the present invention is applied to the deformation analysis of functionally graded piezoelectric hollow cylinders.

[0149] Research is conducted on a functionally graded piezoelectric hollow cylinder, such as Figure 1 (b) As shown. Through the functional gradient piezoelectric hollow cylinder optimization design scheme of the present invention, the functional gradient piezoelectric hollow cylinder is optimized. Figure 1 As shown in (b), the ratio of the inner diameter to the outer diameter of the functionally graded piezoelectric hollow cylinder is taken as R i =380mm, R o =620mm, the mechanical load on the inner surface of the structure p = 0.1MPa, and the electrical load V = 0. The physical information network adopts a 30-layer hidden layer structure (30 neurons per layer), uses Tanh as the activation function, and sets the learning rate to 1×10 -5 , after 2×10 4 The deep regression network is configured with 20 hidden layers (64 neurons per layer), using the ReLU activation function, and 5 The mathematical combination of coordinate points, constants, and operators is optimized during this training. To ensure the engineering practicality of the expression, the maximum length is set to 16 tokens, the nesting of trigonometric functions is limited to no more than two layers, and the nesting depth of exponential / logarithmic operations is strictly controlled. Through dual-network collaborative optimization, the explicit expression of volume fraction that can be directly used in engineering implementation is finally obtained:

[0150]

[0151] in It is a Sigmoid function, and at the same time the parameter k in the volume fraction function expression is determined to be 1.5.

[0152] Table 1 PZT-4 and PZT-5H material parameters

[0153]

[0154]

[0155] Based on the volume fraction obtained through network training, the displacement and stress components can be further calculated. The results are shown in the attached figure. Figure 3 The volume fraction function expression of the functionally gradient material obtained based on network training can be further combined with finite element analysis to calculate the stress and strain distribution of the material under different loads, thereby conducting quantitative analysis of the elastic deformation mechanics of the material under different loads.

[0156] The proposed method combines deep regression with physical information network training to directly obtain an explicit expression for the optimal volume fraction function in functionally graded materials (FGMs), while also determining the parameters within the expression. Specifically, the volume fraction function expression for the optimized design of FGMs is derived entirely through neural network training, significantly reducing computational costs. This provides a more scientific and systematic solution for the optimized design of FGMs and offers technical support for their engineering applications.

Claims

1. A functional gradient piezoelectric material optimization method based on physical information network and deep regression, characterized in that: The following steps are included: Step 1: Construct a physical information neural network embedded in the piezoelectric constitutive relationship. The physical information neural network takes spatial coordinates and material parameters as input variables, outputs displacement field, stress field, electric potential distribution, and key parameters of the piezoelectric phase volume fraction, and uses automatic differentiation technology to accurately calculate the stress field and electric displacement field. By integrating the mechanical equilibrium equations, electrical control equations, boundary conditions, and physical constraints of the optimization objective into the loss function, it ensures that the network prediction results strictly conform to the mechanical-electrical coupling characteristics of the piezoelectric material and achieve the optimization goal. Step 2: Develop a deep regression network based on symbolic regression to convert the piezoelectric phase volume fraction distribution predicted by the neural network into an analyzable mathematical expression; Step 3: Use an alternating iterative strategy to jointly optimize the two networks; while keeping the parameters of the deep regression network unchanged, prioritize training the physical information neural network to meet the accuracy requirements of the displacement field, stress field, electric potential distribution, and piezoelectric phase volume fraction; then fix the physical information network, optimize the expression generation capability of the deep regression network, and finally resolve the piezoelectric phase volume fraction expression.

2. The functional gradient piezoelectric material optimization method based on physical information network and deep regression according to claim 1 is characterized in that: The step 1 is specifically as follows: Build a fully connected physical information neural network W net , the input is the constitutive parameters of the functionally graded piezoelectric material (such as the elastic modulus coefficient c ijkl , piezoelectric constant e ijk , dielectric constant κ ij ) and structural coordinate position points (x, y, z), the output is the displacement distribution (u, v, w), the potential distribution φ and the volume fraction function V A ,Right now {(u,v,w) pred ,ϕ pred (x,y,z),V Apred (x,y,z)}=W net (x,y,z;ω,b) (1) where u pred (x,y,z),v pred (x,y,z),w pred (x,y,z),φ pred (x,y,z) and V Apred (x, y, z) are the displacement distribution, potential distribution, and volume fraction function predicted based on the coordinate position (x, y, z), ω represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network. The network realizes the nonlinear transformation from the input space to the multi-physical field response through the parameterized mapping of the weight ω and the bias b.

3. The functionally graded piezoelectric material optimization method based on physical information network and deep regression according to claim 2 is characterized in that: The constitutive parameters of the functionally graded piezoelectric material are expressed as a function of volume fraction, and the equivalent material parameters are calculated by a homogenization model; The functionally graded piezoelectric material is composed of component A (PZT-4) and component B (PZT-5H), and the volume fractions of component A and component B are V A (x,y,z) and V B (x,y,z), satisfying V A (x,y,z)+V B (x, y, z) = 1, using the Voigt mixing model, the equivalent material parameters are expressed as in and are the material parameters of the piezoelectric material PZT-4 (PZT-5H); Based on the displacement and potential distribution output by the network, the strain and electric field intensity of the functionally graded piezoelectric structure are obtained by using the automatic differentiation technique. E i =-φ ,i (4) where ε ij is the strain, u i is the displacement component, E i is the electric field intensity component, φ is the electric potential component, and the subscript "," indicates the derivative; Combined material equivalent parameter c ijkl 、e ijk , κ ij and the above strain ε ij and the electric field strength E i , using the constitutive equation, the stress and electric displacement components of the functionally graded piezoelectric structure are obtained as where σ ij is the stress component, D i is the electric displacement component.

4. The functionally graded piezoelectric material optimization method based on physical information network and deep regression according to claim 3 is characterized in that: The control equation of the equilibrium equation of the functionally graded piezoelectric structure is composed of the mechanical equilibrium equation and the electrical Gaussian law, namely: in is the body force component; In the definition of the boundary Γ of the functionally graded piezoelectric structure, there are two boundary conditions: mechanical or electrical, namely in For a given displacement, Γ u is the displacement boundary of the piezoelectric structure, For a given surface tension, Γ σ is the stress boundary of the piezoelectric structure.

5. The functionally graded piezoelectric material optimization method based on physical information network and deep regression according to claim 4 is characterized in that: The electrical boundary conditions are in For a given electric displacement, Γ φ is the potential boundary of the piezoelectric structure, For a given charge, Γ q is the electric displacement boundary. By substituting the constitutive equation into the equilibrium equation, the control equation with displacement and potential as the basic unknowns is established. Combined with the boundary conditions, the displacement, stress and potential distributions of the functionally graded piezoelectric structure are obtained. For functionally graded piezoelectric structures, the optimization objective is defined as in Or the optimization objective is defined as f M =max(φ output (x,y,z)) (11).

6. The functionally graded piezoelectric material optimization method based on physical information network and deep regression according to claim 5, characterized in that: The network training process adopts a multi-objective optimization strategy. The total loss function Loss consists of three parts: the mean square error Loss of the balance equation f , boundary condition mean square error Loss b And the objective function mean square error Loss M ,Right now Loss=Loss f +Loss b +Loss M (12) Mean square error Loss of functional gradient structural equilibrium equation f The mean square error of the boundary conditions of the functionally graded piezoelectric structure is Loss b for Objective function mean square error Loss M for Loss M =s M or M (13c) The residual terms of the equilibrium equations are calculated by automatic differentiation to ensure that the physical laws are strictly satisfied; the boundary condition terms force the network prediction to meet the given constraints.

7. The functionally graded piezoelectric material optimization method based on physical information network and deep regression according to claim 6, characterized in that: The step 2 is specifically as follows: A deep regression network was constructed, which used a symbolic modeling method to establish an explicit mathematical relationship between spatial coordinates and the volume fraction of the piezoelectric phase. First, an expression generator based on a binary tree structure was established, with leaf nodes set as input variables (spatial coordinates x, y, z and parameters to be optimized c1-c n ), non-leaf nodes are configured as operator nodes (including the following basic arithmetic operators), namely 8. The functionally graded piezoelectric material optimization method based on physical information network and deep regression according to claim 7, characterized in that: The network optimization adopts a two-stage hybrid algorithm: the first stage implements genetic algorithm optimization; the second stage switches to gradient descent optimization, using adaptive learning rate to fine-tune the leaf node parameters; after each round of iteration, the candidate expression is double-evaluated: the normalized fitness index R = 1 / (1 + NRMSE) based on the output data of the physical information network is calculated, where NRMSE is the normalized root mean square error, which is defined as the root mean square of the deviation between the predicted value and the target value normalized by the data variance. Where S is the variance of the volume fraction data set calculated by the target volume fraction function, V A (x,y,z) and V A * (x, y, z) are the volume fraction function expressions predicted by the physical information network and the deep regression network, respectively; In the expression screening stage, a multi-objective optimization mechanism is established, and three key constraints are set: fitness index R ≥ 0.93, total number of nodes ≤ 35, and function depth ≤ 7. The physical rationality of the expressions that pass the initial screening is verified, including dimensional consistency check (forcing all operations to remain dimensionless or with the correct dimension) and boundary condition verification. Finally, the optimal solution set of the expression is generated in the output stage.

9. The functionally graded piezoelectric material optimization method based on physical information network and deep regression according to claim 8, characterized in that: The step three is specifically as follows: The intelligent optimization of functionally graded piezoelectric materials is achieved through the collaborative work of physical information neural networks and deep regression networks. Two independent networks are configured in the initialization phase: The physical information neural network adopts a 4-layer fully connected architecture. The input layer receives spatial coordinates and process parameters, and the output layer generates displacement fields, electric potential fields, and constitutive parameters. The network embeds the piezoelectric constitutive equation as a hard constraint. The deep regression network is constructed as a symbolic regression network, which expresses the component distribution function based on a binary tree structure, and the node library contains basic operations.

10. The functionally graded piezoelectric material optimization method based on physical information network and deep regression according to claim 9, characterized in that: The training process implements an intelligent alternating optimization strategy: First, the parameters of the deep regression network are fixed, and the adaptive momentum optimizer is used to train the physical information neural network. The loss function includes the mechanical balance term, the electrical Gaussian law term, and the boundary conditions. The automatic differentiation technique is used to accurately calculate the stress field and the electric displacement field gradient. Then, the parameters of the physical information neural network are frozen, and the hybrid optimization algorithm of the deep regression network is enabled. Combining the genetic algorithm and the gradient descent method, the two indicators are calculated in real time during the optimization process: the goodness of fit R 2 (target > 0.99) and expression complexity (number of nodes ≤ 30); The system sets a dynamic termination mechanism: monitoring the composite loss value of the physical information neural network (threshold 1×10 -5 ) and the normalized fitness R (threshold 0.99) of the deep regression network. Early stopping is triggered when the two indicators meet the standards continuously. After multiple rounds of iterative optimization, the system finally outputs an explicit mathematical expression with clear physical meaning.

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